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The weight hierarchy of a linear [n, k;q] code C over GF(q) is the sequence (d1,d2,..., dk) where dr is the size of the smallest support of an r-dimensional subcode of C. An [n,k;q] code satisfies the chain condition if there exists subcodes D1 D2 … Dk = C of C such that Dr has dimension r and support of size dr for all r. Further, C satisfies the almost chain condition if it does not satisfy the chain condition, but there exist subcodes Dr of dimension r and support of size dr for all r such that D2 D3 Dk = C and D1 D3. A simple necessary condition for a sequence to be the weight hierarchy of a code satisfying the almost chain condition is given. Further, explicit constructions of such codes are given, showing that in almost all cases, the necessary conditions are also sufficient.
The weight hierarchy of a linear [n, k; q] code C over GF (q) is the sequence (d1, d2, ..., dk) where dr is the size of the smallest support of an r-dimensional subcode of C. An [n, k; q] code satisfies the chain condition if there exists a subcategory D1 D2 ... Dk = C of C such that Dr has dimension r and support of size dr for all r. Further, C satisfies the almost chain condition if it does not satisfy the chain condition, but there exist subcodes Dr of dimension r and support of size dr for all r such that D2 D3 Dk = C and D1 D3. A simple necessary condition for a sequence to be the weight hierarchy of a Further, explicit constructions of such codes are given, showing that in almost all cases, the necessary conditions are also sufficient.